arXiv:astro-ph/9601188v1 31 Jan 1996 aaerzdi em fΩ= Ω of terms in parametrized in scale largest the on Universe. DM the demon- of presence POTENT the and strate IRAS in DM of space. presence intercluster the show distri- also clusters velocity in galaxy bution by The filled halos matter. many large nonluminous for in evidence curves provide Rotation galaxies established. reliably is .Itouto Csooia environ- (Cosmological Introduction 1. o data structures, scale o large context of the spectrum in mass considered observed are the particles These Majoron. and .Berezinsky V. Matter Dark Non–Baryonic 1 1 as h 4 3 2 ueet ugs eeal 0 generally suggest surements peetdb .Berezinsky) V. by (presented rlcniaefrHMi h evetneutrino, heaviest the is nat- HDM The CDM. for – candidate not ural if they particles, relativistic HDM are called crosses particles are horizon If when scale. moment galactic the the at particles of ig n oacutr arwdti nevlto interval 0 this narrowed clusters Coma and in Virgo Cepheids extragalactic of measurements recent loytei sgvn[]a Ω as [1] given is cleosynthesis (CDM). DM cold and (HDM) DM hot DM, measuremets. uncer- difficult to these due in interval involved this tainties of accuracy the about NN aoaoiNzoaidlGa as,600Asri(AQ) Assergi 67010 Sasso, Gran del Nazionali Laboratori INFN, hoeia hsc iiin EN H11 eea2,Switzerlan Italy Torino, 23, Geneva I-10125 CH–1211 1, CERN, Giuria P. Division, via Physics Torino, Theoretical Italy di Torino, Sezione I-10125 - 1, INFN Giuria P. via Torino, Universit`a di . . 88 6 h atrdniyi h Universe the in density matter The Universe the in (DM) matter dark of Presence h etpril addtsfrnnbroi oddr mat dark cold non–baryonic for candidates particle best The o n odD r itnuse yvelocity by distinguished are DM cold and Hot h est fbroi atrfudfo nu- from found matter baryonic baryonic of in density The subdivided be can Matter Dark stedmninesHbl osatdefined constant Hubble dimensionless the is h ment) ≤ · = 10 h H − ≤ 0 29 / 0 h (100 . .Hwvr n hudb cautious be should one However, 9. 2 1 .Bottino A. , g/cm km.s 3 − steciia est and density critical the is 1 .Mpc 2 , 3 − b h n .Mignola G. and 1 . 4 2 ρ/ρ .Dffrn mea- Different ). 0 = ≤ c where , h . 025 ρ ≤ susually is ± .The 1. 0 ρ . 005 c 3 , ≈ 4 . h euyo h nainr cnro)Ω scenarios) inflationary the and of problem beauty flatness the problem, (horizon tivation Ω give hyas rdc ieeteet o cluster- for effects different produce etc. CfA also IRAS, COBE, They of data These with to compared fluctuations be of energy). spectra vacuum different (baryonic predict and models above CDM described HDM, DM DM, of types four the value. observa- this contradict No by significantly supported data analysis. is tional POTENT value and This data IRAS assumed. usually is n nrydniyi ie yΩ by given is density correspond- energy The ing described Λ. constant energy cosmological the vacuum by the namely Universe, n f.CMpu ayncmte a ex- can density total matter if fluctuations Ω baryonic of spectrum plus the plain CDM CfA. IRAS a COBE, and of has measurements with fluctuations DM compared of as spectrum baryonic the for plus prediction Universe. wrong HDM in DM with of Universe properties the to restrictions candidates. CDM as suggested were naturally most iyi siae sΩ as estimated is sity Ω of terms 0.7. than in the density: restrict energy results vacuum COBE the and lensing Quasar lseso aais g fteUies etc. Universe the of age galaxies, of clusters n omlgclmdl ihtersrcin ie by given restrictions the with models cosmological f 0 hr r eea omlgclmdl ae on based models cosmological several are There hr soemr omo nrydniyi the in density energy of form more one is There h tutr omto nUies u strong put Universe in formation structure The otiuino aatchlst h oa den- total the to halos galactic of Contribution e r eiwd aey etaio xo,axino axion, neutralino, namely, reviewed, are ter ≈ 0 . ≈ 3. 0 Italy , . .Isie otyb hoeia mo- theoretical by mostly Inspired 3. d τ nurn.Mn e particles new Many -neutrino. ∼ 0 . 03 − 0 Λ . n clusters and 1 Λ = Λ ti less is it / (3 0 H 1 = 0 2 ). 1 2
2 cluster correlations, velocity dispersion etc. The also predicts ΩCDM h ≈ 0.15 with uncertainties simplest and most attractive model for a cor- 0.1. Finally, we shall mention that the CDM with rect description of all these phenomena is the Ω0 =ΩCDM =0.3 and h =0.8, which fits the ob- so-called mixed model or cold-hot dark matter servational data, also gives Ωh2 ≈ 0.2. Therefore model (CHDM). This model is characterized by Ωh2 ≈ 0.2 can be considered as the value common following parameters: for most models. In this paper we shall analyze several candi- ΩΛ =0, Ω0 =Ωb +ΩCDM +ΩHDM =1, dates for CDM, best motivated from the point of −1 −1 H0 ≈ 50 kms Mpc (h ≈ 0.5), view of elementary particle physics. The motiva-
ΩCDM :ΩHDM :Ωb ≈ 0.75:0.20:0.05, (1) tions are briefly described below. Neutralino is a natural lightest supersymmetric where ΩHDM ≈ 0.2 is obtained in ref.[2] from particle (LSP) in SUSY. It is stable if R-parity is damped Lyα data. Thus in the CHDM model conserved. Ωχ ∼ ΩCDM is naturally provided by the central value for the CDM density is given by annihilation cross-section in large areas of neu- 2 tralino parameter space. Ω h =0.19 (2) CDM Axion gives the best known solution for strong with uncertaities within 0.1. CP-violation. Ωa ∼ ΩCDM for natural values of The best candidate for the HDM particle is τ- parameters. neutrino. In the CHDM model with Ων = 0.2 Axino is a supersymmetric partner of axion. It mass of τ neutrino is mντ ≈ 4.7 eV . This com- can be LSP. ponent will not be discussed further. Majoron is a Goldstone particle in spontaneously The most plausible candidate for the CDM par- broken global U(1)B−L or U(1)L. KeV mass can ticle is probably the neutralino (χ): it is massive, be naturally produced by gravitational interac- stable (when the neutralino is the lightest super- tion. symmeric particle and if R-parity is conserved) Apart from cosmological acceptance of DM and the χχ-annihilation cross-section results in particles, there can be observational confirma- 2 Ωχh ∼ 0.2 in large areas of the neutralino pa- tion of their existence. The DM particles can rameter space. be searched for in the direct and indirect experi- In the light of recent measurements of the Hub- ments. The direct search implies the interaction ble constant the CHDM model faces the age prob- of DM particles occurring inside appropriate de- lem. The lower limit on the age of Universe tectors. Indirect search is based on detection of t0 > 13 Gyr (age of globular clusters) imposes the the secondary particles produced by DM particles upper limit on the Hubble constant in the CHDM in our Galaxy or outside. As examples we can −1 −1 model H0 < 50 kms Mpc . This value is in mention production of antiprotons and positrons slight contradiction with the recent observations in our Galaxy and high energy gamma and neu- of extragalactic Cepheids, which can be summa- trino radiation due to annihilation of DM parti- −1 −1 rized as H0 > 60 kms Mpc . However, it is cles or due to their decays. too early to speak about a serious conflict tak- ing into account the many uncertainties and the 2. Axion physical possibilities (e.g. the Universe can be locally overdense - see the discussion in ref.[3]). The axion is generically a light pseudoscalar The age problem, if to take it seriously, can particle which gives natural and beautiful solu- be solved with help of another successful cosmo- tion to the CP violation in the strong interaction logical model ΛCDM. This model assumes that [4] (for a review and references see[5]). Sponta- Ω0 = 1 is provided by the vacuum energy (cos- neous breaking of the PQ-symmetry due to VEV mological constant Λ) and CDM. From the limit of the scalar field <φ>= fP Q results in the ΩΛ < 0.7 and the age of Universe one obtains production of massless Goldstone boson. Though ΩCDM ≥ 0.3 and h < 0.7. Thus this model fP Q is a free parameter, in practical applications 3
10 12 it is assumed to be large, fP Q ∼ 10 − 10 GeV stars. The upper limit from SN 1987A was re- and therefore the PQ-phase transition occurs in considered taking into account the nucleon spin very early Universe. At low temperature T ∼ fluctuation in N +N → N +N +a axion emission. ΛQCD ∼ 0.1 GeV the chiral anomaly of QCD There are three known mechanisms of cosmo- induces the mass of the Goldstone boson ma ∼ logical production of axions. They are (i)thermal 2 ΛQCD/fP Q . This massive Goldstone particle is production, (i) misalignment production and (iii) the axion. The interaction of axion is basically radiation from axionic strings. determined by the Yukawa interactions of field(s) The relic density of thermally produced axions φ with fermions. Triangular anomaly, which pro- is about the same as for light neutrinos and thus −2 vides the axion mass, results in the coupling of for the mass of axion ma ∼ 10 eV this compo- the axion with two photons. Thus, the basic for nent is not important as DM. cosmology and astrophysics axion interactions are The misalignment production is clearly ex- those with nucleons, electrons and photons. plained in ref.[5]. Numerically, axion mass is given by At very low temperature T ≪ ΛQCD the mas- sive axion provides the minimum of the potential −3 10 ma =1.9 · 10 (N/3)(10 GeV/fP Q) eV, (3) at value θ = 0,which corresponds to conservation of CP. At very high temperatures T ≫ ΛQCD where N is a color anomaly (number of quark the axion is massless and the potential does not doublets). depend on θ. At these temperatures there is no All coupling constants of the axion are inversely reason for θ to be zero: its values are different in proportional to fP Q and thus are determined by various casually disconnected regions of the Uni- the axion mass. Therefore, the upper limits on verse. When T → ΛQCD the system tends to go emission of axions by stars result in upper limits to potential minimum (at θ = 0) and as a re- for the axion mass. Of course, axion fluxes can- sult oscillates around this position. The energy not be detected directly, but they produce addi- of these coherent oscillations is the axion energy tional cooling which is limited by some observa- density in the Universe. From cosmological point tions (e.g.,age of a star, duration of neutrino pulse of view axions in this regime are equivalent to in case of SN etc). In Table 1 we cite the upper CDM. The energy density of this component is limits on axion mass from ref.[5], compared with approximately [5,7] revised limits, given recently by Raffelt [6]. 2 −5 −1.18 Ωah ≈ 2 · (ma/10 eV ) . (4) Uncertainties of the calculations can be estimated Table 1 as 10±0.5. Astrophysical upper limits on axion mass Axions can be also produced by radiation of axionic strings [5],[8]. Axionic string is a one- 1990 [5] 1995 [6] dimension vacuum defect < φP Q >= 0, i.e. a sun 1 eV 1 eV line of old vacuum embedded into the new one. very The string network includes the long strings and red giants 1 · 10−2 eV uncertain closed loops which radiate axions due to oscilla- hor.–branch not tion. There were many uncertainties in the axion 0.4 eV stars considered radiation by axionic strings (see ref.[5] for a re- SN 1987A 1 · 10−3 eV 1 · 10−2 eV view). Recently more detailed and accurate cal- culations were performed by Battye and Schellard [8]. They obtained for the density of axions Ω h2 ≈ A(m /10−5 eV )−1.18 (5) As one can see from the Table the strong upper a a limit, given in 1990 from red giants, is replaced with A limited between 2.7 and 15.2 and with by the weaker limit due to the horizontal-branch uncertainties of the order 10±0.6. The overpro- 4
2 duction condition Ωah > 1 imposes lower limit two gluons). The axino can be the lightest super- −5 on axion mass ma > 2.3 · 10 eV . symmetric particle and thus another candidate 2 Fig.1 shows the density of axions Ωah as a for DM. function of the axion mass ma. The upper limits How heavy the axino can be? The mass of ax- on axion mass from Table I are shown above the ino has a very model dependent value. In the phe- upper absciss (limits of 1990) and below lower ab- nomenological approach, using the global super- sciss (limits of 1995). The overproduction region symmetry breaking parameter MSUSY one typi- 2 Ωah > 1 and the regions excluded by astrophysi- cally obtains (e.g. [11],[12]) cal observations [6] are shown as the dotted areas. 2 m˜ ∼ M /f (6) The axion window of 1995 (shown as undot- a SUSY P Q ted region) became wider and moved to the right For example, if global SUSY breaking occurs due as compared with window 1990. The horizontal to VEV of auxiliary field of the goldstino su- strip shows ΩCDM =0.2 ± 0.1 as it was discussed permultiplet Figure 1. Axion window 1995. The curves ”therm.” and ”misalign.” describe the thermal and misaligne- ment production of axions, respectively. The dash-dotted curve corresponds to the calculations by Davis [10] for string production. The recent refined calculations [8] are shown by two dashed lines for two extreme cases, respectively. The other explanations are given in the text. Since axino interacts with matter very weakly, If the axino is LSP and the neutralino is the the decoupling temperature for the thermal pro- second lightest supersymmetric particle, the ax- duction is very high [16]: inos can also be produced by neutralino decays 9 11 [15],[16],[17]. According to estimates of ref.[17] Td ≈ 10 GeV (fP Q/10 GeV ). (7) the axinos are produced due to χ → a˜ + γ de- 8 Therefore, axinos are produced thermally at the cays at the epoch with red-shift zdec ∼ 10 . Ax- reheating phase after inflation. The relic concen- inos are produced in these decays as ultrarela- tration of axinos can be easily evaluated for the tivistic particles and the free-streeming prevents reheating temperature TR as the growth of fluctuations on the horizon scale 10 4 2 ma˜ 3 · 10 GeV 2 TR and less. At red-shift znr ∼ 10 axinos become Ωa˜h ≈ 0.6 ( ) 9 (8) 100 keV fP Q 10 GeV non-relativistic due to adiabatic expansion (red 9 shift). From this moment on the axinos behave as Reheating temperature TR ≤ 10 GeV gives no the usual CDM and the fluctuations on the scales problem with the gravitino production. The relic 2 λ ≥ (1 + znr)ctnr (which correspond to a mass density (8) provides ΩCDM h ∼ 0.2 for a reason- 15 larger than 10 M⊙) grow as in the case of stan- able set of parameters ma˜,fP Q and TR. One can dard CDM. For smaller scales the fluctuations, easily incorporate in these calculations the addi- as was explained above, grow less than in CDM tional entropy production if it occurs at EW scale model. Therefore, as was observed in ref.[17], the [17]. 6 axinos produced by neutralino decay behave like Another interesting possibility was considered HDM. It means that axinos can provide generi- recently in ref.[22]. In this model the Majoron cally both components, CDM and HDM, needed is rather strongly coupled with νµ and ντ neu- for description of observed spectrum of fluctua- trinos (Jνµντ coupling). The Majorons are pro- tions. duced through ντ → J + νµ decays. The strong Unfortunatelly stable axino is unobservable. In interactions between Majorons reduces the relic case of very weak R-parity violation, decay of ax- abundance of the Majorons to the cosmologically inos can produce a diffuse X-ray radiation, with required value. practically no signature of the axino. The Majoron signature in all these models is given by J → γ + γ decays, which result in the production of keV X-ray line in the X-ray back- 4. Majoron ground radiation. The Majoron is a Goldstone particle associ- ated with spontaneously broken global U(1)B−L 5. Neutralino or U(1) symmetry. The symmetry breaking oc- L The neutralino is a superposition of four spin curs due to VEV of scalar field, <σ>= vs, and 1/2 neutral fields: the wino W˜ 3, bino B˜ and two σ splits into two fields, ρ and J: Higgsinos H˜1 and H˜2: σ → (vs + ρ) exp(iJ/vs). (9) χ = C1W˜ 3 + C2B˜ + C3H˜1 + C4H˜2 (10) The field J is the Majoron. A mass of ∼ keV The neutralino is a Majorana particle. With a can be obtained due to gravitational interac- unitary relation between the coefficients Ci the tion[18],[19]. The keV Majoron has the great parameter space of neutralino states is described cosmological interest since the Jeans mass associ- by three independent parameters, e.g. mass of 3 2 ated with this particle, mJeans ∼ mPl/mJ , gives wino M2, mixing parameter of two Higgsinos µ, 12 the galactic scale M ∼ 10 M⊙. In all other re- and the ratio of two vacuum expectation values spects the keV Majoron plays the role of a CDM tan β = v2/v1. particle. It is assumed usually that the Majoron In literature one can find two extreme ap- interacts directly only with some very heavy par- proaches describing the neutralino as a DM par- ticles (e.g. with the right-handed neutrino νR). ticle. It results in very weak interaction of the Majoron (i)Phenomenological approach. The allowed with the ordinary particles (leptons, quarks etc) neutralino parameter space is restricted by the and thus makes the Majoron ”invisible” in the LEP and CDF data. In particular these data accelerator experiments. put a lower limit to the neutralino mass, mχ > The cosmological production of the Majoron 20 GeV. In this approach only the usual GUT re- occurs through thermal production[19,20] and ra- lation between gaugino masses, M1 : M2 : M3 = diation by the strings [19] as in case of the ax- α1 : α2 : α3, is used as an additional assumption, ion. However, under imposed observational con- where αi are the gauge coupling constants. All straints, the Majoron in models[18],[19] has to be other SUSY masses which are needed for the cal- in general unstable with lifetime much shorter culations are treated as free parameters, limited than the age of Universe. A successful model from below by accelerator data. was developed in ref.[21], where the Majoron was One can find the relevant calculations within assumed to interact with the ordinary particles this approach in refs.[23,24] and in the review[25] through new heavy particles. For the cosmologi- (see also the references therein). There are large cal production it was considered the phase transi- areas in neutralino parameter space where the tion associated with the global U(1)B−L symme- neutralino relic density satisfies the relation (2). try breaking, when the Majorons were produced This is especially true for heavy neutralinos with both directly and through ρ → J + J decays. mχ > 100 − 1000 GeV, ref.[26]. In these areas 7 there are good prospects for indirect detection of ref.[36]). The powerful restriction from the no- neutralinos, due to high energy neutrino radia- fine-tuning condition is added. tion from Earth and Sun (see [27,28] and refer- ences therein) as well as due to production of an- 5.1. SUSY theoretical framework tiprotons and positrons in our Galaxy. The direct The basic element which should be used in detection of neutralinos is possible too, though the analysis is a supersymmetry breaking and in- in more restricted parameter space areas of light duced by it (through radiative corrections) elec- neutralinos (see review [25]). troweak symmetry breaking [38]. We shall refer This model-independent approach is very inter- to this restriction as to the EWSB restriction. esting as an extreme case: in the absence of an ex- One starts with unbroken supersymmetric perimentally confirmed SUSY model it gives the model described by some superpotential. It is results obtained within most general framework assumed that local supersymmetry is broken by of supersymmetric theory. supergravity in the hidden sector, which commu- (ii) Strongly constrained models. This approach nicates with the visible sector only gravitation- is based on the remarkable observation that in ally. This symmetry breaking penetrates into the the minimal SUSY SU(5) model with fixed parti- visible sector in the form of global supersymme- cle content, the three running coupling constants try breaking. More specifically it is assumed that meet at one point corresponding to the GUT mass the symmetry breaking terms in the visible sec- MGUT . Because of the fixed particle content tor are the soft breaking terms given at the GUT 2 2 of the model, its predictions are rigid and they scale Q ∼ MGUT by the following expression: strongly restrict the neutralino parameter space. 2 2 This is especially true for the limits due to pro- Lsb = m0 X |φa| + m1/2 X λaλa + + ton decay p → K ν. As a result very little space a a is left for neutralino as DM particle. Normally + Am0fY + Bm0µH1H2 (11) neutralinos overclose the Universe (Ωχ > 1). The relic density decreases to the allowed values in where φa are scalar fields of the model (sfermions very restricted areas where χχ-annihilation is ac- and two Higgses H1 and H2), λa are gaug- 0 cidentally large (e.g.due to the Z exchange term ino fields, fY are trilinear Yukawa couplings of - see ref.[29]. Thus, this approach looks rather fermions and Higgses and the last term is an pessimistic for neutralino as DM particle. additional (relative to the superpotential term In several recent works [30]-[35] less restricted µH1H2) soft breaking mixing of two Higgses. SUSY models were considered. In particular the Here and everywhere below we specify the scale at limits due to proton decay were lifted. A GUT which an expression and parameters are defined. model was not specified or less restrictive SO(10) The soft breaking terms (11) are described by model was used [35]. Although, the neutralino 5 free parameters : m0,m1/2,A,B and µ. This can be heavy in these models, the prospects for implies the strong assumption that all scalars φa indirect detection, including the detection of high and all gauginos λa at the GUT scale have the energy neutrinos from the Sun and Earth, are common masses m0 and m1/2, respectively. This rather pessimistic [32]. The direct detection is assumption can be relaxed, as we shall discuss possible in many cases [33,34,32]. later. (iii)Relaxed restrictions. In section 5.3, follow- The soft breaking terms (11) together with su- ing refs.[36],[37], we shall analyze the restrictions persymmetric mixing give the following potential to neutralino as DM particle, imposed by basic defined at the EW scale at the tree level: properties of SUSY theory. As in many pre- 2 2 2 2 vious works, a fundamental element of analysis V = m1|H1| + m2|H1| − is the radiatively induced EW symmetry break- − Bµm0(H1H2 + hc)+ ing (EWSB)[38]. However, some mass unifica- 2 2 g1 + g2 2 2 2 + (|H1| − |H2| ) , (12) tion conditions at the GUT scale are relaxed (see 8 8 The mass parameters m1 and m2 at the GUT where Ji are also numerical coefficients. scale are equal to Eq.(16) allows to impose the no-fine-tuning condition in the neutralino parameter space. In- m2(GUT )= m2(GUT )= µ2 + m2, (13) 1 2 0 deed, one can keep large values of masses in the with µ defined at the GUT scale,too. The term µ2 rhs of Eq(16) only by the price of accidental com- in Eqs.(12),(13) appears due to the mixing term pensation between the different terms. It is un- µH1H2 in the superpotential of unbroken SUSY. natural to expect an accidental compensation to The radiative EWSB occurs due to evolution a value less than 1% from the initial values. This of mH2 , the mass of H2, which is connected with is the no-fine-tuning condition. the upper components of the fermion doublets Naturally this condition is just the same as the and in particular with t-quark. Because of the one due to the radiative corrections to the Higgs large mass of t-quark and consequently the large mass. 2 2 Yukawa coupling YttH2 , mH2 evolves from m0 at 2 5.2. Restrictions: the price list the GUT scale to the negative value m < 0 at H2 Within the theoretical framework outlined the EW scale. At this value the potential (12) above one can choose the restrictions from the acquires its minimum and the system undergoes following price list: the EW phase transition coming to the minimum of the potential. At EW scale in the tree approx- • Soft breaking terms (11) and EWSB condi- imation the conditions of the potential minimum tions (14), (vanishing of the derivatives) give: • No-fine-tuning condition, m2 − m2 tan2 β M 2 µ2 = H1 H2 − Z tan2 β − 1 2 • Particle phenomenology (constraints from −2Bµ accelerator experiments and the condition sin 2β = 2 2 2 (14) that the neutralino is the LSP), mH1 + mH2 +2µ With these equations we obtain one connec- • Restrictions due to b → sγ decay, tion between five free parameters describing the soft-breaking terms (11). Thus the number • Meeting of coupling constants at MGUT , of independent parameters is reduced to four, • b − τ and b − τ − t unification, e.g.m0,m1/2,A,µ (or tan β). Using the renormalization group equations • Restrictions due to p → Kν decay. (RGE) one can follow the evolution of the scalar particles (Higgses and sfermions) and spin 1/2 Using some (or all) restrictions listed above one particles (gauginos) from the masses m0 and m1/2 can start calculations for the neutralino as DM at the GUT scale to the masses at the EW particle. The regions where the neutralinos are 2 scale. Analogously, the evolution of the coupling overproduced (Ωχh > 1) must be excluded from constants can be calculated. In particlular the consideration and the allowed region should be masses of Higgses at EW scale are given by determined according to the chosen cosmological 2 2 2 2 model (e.g. Ωχh = 0.2 ± 0.1 for the CHDM mHi = aim0 + bim1/2 + model). For the allowed regions the signal for 2 2 + ciA m0 + diAm0m1/2, (15) direct and indirect detection can be calculated. where ai,bi,ci and di are numerical coefficients, 5.3. SUSY models with basic restrictions which depend on tan β. Accepting all restrictions listed above one ar- Equivalently, using Eqs.(14) one finds rives at a rigid SUSY model, with the neutralino 2 2 2 2 2 parameter space being too strongly constrained. M = J1m + J2m + J3A m + Z 1/2 0 0 In ref.[37] the SUSY models with basic restric- 2 + J4Am0m1/2 − µ , (16) tions were considered. These restrictions are as 9 follows: (i)Radiative EWSB, (ii) No fine-tuning stronger than 1%, (iii) RGE and particle phenomenology (accelerator limits on the calculated masses and the condition that neutralino is LSP), (iv) Lim- its from b → sγ decay taken with the uncertain- ties in the calculations of the decay rate and (v) 2 0.01 < Ωχh < 1 as the allowed relic density for neutralinos. Rather strong restrictions are im- posed by the condition (ii); in particular it limits the mass of neutralino as mχ < 200 GeV. At the same time some restrictions are lifted as being too model-dependent: (i) No restrictions are imposed due to p → Kν decay, (ii) Unifica- tion of coupling constants at the GUT point is allowed to be not exact (it is assumed that new very heavy particles can restore the unification), (iii) unification in the soft breaking terms (11) is relaxed. Following ref.[36] it is assumed that masses of Higgses at the GUT scale can deviate from the universal value m0 as 2 2 Figure 2. The neutralino parameter space for the mHi (GUT )= m0(1 + δi) (i =1, 2). (17) mass–unification case δ1 = δ2 = 0 and tan β = 8. This non-universality affects rather strongly the properties of neutralino as DM particle: the allowed parameter space regions become small boxes. As one can see in most regions the larger and neutralino is allowed to be Higgsino- neutralinos are overproduced. The allowed re- dominated, which is favorable for detection. gions correspond to large χχ annihilation cross- Some results obtained in ref.[37] are illustrated section (e.g. due to Z0-pole). by Figs. 2 - 6. The regions allowed for the neu- Fig. 3 and Fig. 2 differ only by universality: in tralino as CDM particle are shown everywhere by Fig. 3 δ1 = δ2 = 0 (mass–unification), while in small boxes. Fig. 4 δ1 = −0.2 and δ2 =0.4. The allowed region In Fig.2 the regions excluded by the LEP and in Fig. 3 becomes much larger and is shifted into CDF data are shown by dots and labelled as LEP. the Higgsino dominated region. Figs. 4 and 5 are The regions labelled ”fine tuning” have an acci- given for tanβ = 53. This large value of tan β dental compensation stronger than 1% and thus correspond to b − τ − t unification of the Yukawa are excluded. No-fine-tuning region inside the coupling constants. Again one can notice that broken-line box corresponds to a neutralino mass in the mass–unification case (δ1 = δ2 = 0) only mχ ≤ 200 GeV. The region ”EWSB+particle a small area is allowed for neutralino as CDM phenom.” is excluded by the EWSB condition particle, while in non-universal case (δ1 =0,δ2 = combined with particle phenomenology (neu- −0.2) the allowed area becomes larger and shifts tralino as LSP, limits on the masses of SUSY into the gaugino dominated region. particles etc). In the region marked by rar- In Fig. 6 the scatter plot for the rate of direct efied dotted lines neutralinos overclose the Uni- detection with the Ge detector [39] is given for 2 verse (Ωχh > 1). The solid line corresponds to the non-universal case (δ1 = 0,δ2 = −0.2) and m0 = 0. The regions allowed for neutralino as tan β = 53. We notice that, for some configura- 2 CDM particle (0.01 < Ωχh < 1) are shown by tions, the experimental sensitivity is already at 10 Figure 3. Case δ1 = −0.2, δ2 =0.4 and tan β = 8. Figure 4. Mass–unification case and tan β = 53 the level of the predicted rate. Galaxy and high energy neutrinos from the Sun and Earth). 6. Conclusions The other extreme case, the complete SUSY SU(5) model with fixed particle content, with 1. The density of CDM needed for most cosmo- meeting of coupling constants at the GUT point 2 logical models is given by ΩCDM h = 0.2 ± 0.1. and with the constraints due to proton decay, There are four candidates for CDM, best moti- leaves very little space for the neutralino as DM vated from point of view of elementary particle particle. physics: neutralino, axion, axino and majoron. In the third option the SUSY soft breaking 2. There are different approaches to study the terms (11) and induced EW symmetry breaking neutralino as DM particle in SUSY models with are used as the general theoretical framework. R-parity conservation. Combined with a no-fine-tuning condition this In the phenomenological approach, apart from framework already gives essential restrictions. In the LEP-CDF limits, very few other constraints particular, fine tuning allowed at the level larger are imposed. In the Minimal Supersymmetric than 1% results in the neutralino being lighter Model only one GUT relation between gauginos than 200 GeV. Here the neutralino is gaugino masses, M1 : M2 : M3 = α1 : α2 : α3, is used. dominated, which is unfavorable for direct detec- While the coupling constants are known, the mass tion. If we employ further other restrictions, such parameters needed for calculations are taken as as exact unification of gauge coupling constants the free parameters. Many allowed configurations and soft–breaking parameters at the GUT scale, in the parameter space give the neutralino as DM b → sγ limit and b − τ unification, the model particle with observable signals for direct and in- becomes as rigid as the one considered above. direct detection (antiprotons and positrons in our If, on the other hand, we relax some con- 11 Figure 6. Rate for direct detection (δ1 = 0,δ2 = Figure 5. Case δ1 = 0, δ2 = −0.2 and tan β = 53. −0.2 and tan β = 53). straints, e.g. by assuming non-universality of the scalar mass term in Eq.(11), then even in the case SUSY breaking provides its mass within the in- of 1% no-fine-tuning condition the neutralino can terval 10 − 100 keV . The axino can provide both be the DM particle in a large area of the param- CDM and HDM components needed to fit the cos- eter space and can be detected in some parts of mological observations. The axion can be directly this area in direct and indirect experiments. observed (e.g. in microwave cavity experiments) The neutralino is an observable particle. It can while the axino dark matter is practically unob- be observed directly in the underground experi- servable. ments, or indirectly, mostly due to the products 4. The Majoron with keV mass can be the of neutralino-neutralino annihilation. warm DM particle which explains the galactic 12 3. In the framework of supersymmetric the- scale (M ∼ 10 M⊙) in the structure formation ory, the PQ-mechanism for solving the problem of problem. The decay of the Majoron to two pho- strong CP violation, results in a Goldstone super- tons can produce an observable X-ray line in the multiplet which contains axion and its fermionic cosmic background radiation. superpartner, axino. Both of them can be CDM Acknowledgements particles. The most important parameter here is The main results presented here on the neutralino the scale of PQ symmetry breaking fP Q which are based on a work carried out with John Ellis, is observationally constrained as 2 · 109 GeV < Nicolao Fornengo and Stefano Scopel. We wish to 11 fP Q < 8 · 10 GeV . The axion can be the CDM express our thanks to them for collaboration and particle, if its mass is 10−5 − 10−3 eV ( the corre- discussions. Partial financial support was pro- 9 sponding values of fP Q are between 2 · 10 GeV vided by the Theoretical Astroparticle Network 10 and 6 · 10 GeV ). For larger values of fP Q the under contract No. CHRX–CT93–0120 of the Di- axino can be CDM particle if a mechanism of rection General of the EEC. 12 REFERENCES 22. A.D. Dolgov, S. Pastor and J.W.F. Valle, preprint FTUV–95–14 (1995) and astro– 1. N. Hata, R. Schrerer, G. Steigman, D. ph/9506011. Thomas and T.P. Walker, preprint OSU-TA- 23. A. Bottino, V. de Alfaro, N. Fornengo, G. 26/94 (astro-ph/9412087),1994 Mignola and S. Scopel, Astroparticle Physics 2. A. Klypin, S. Borgani, J. Holtzman and J. 2 (1994) 77. Primack, Ap.J. 444 (1995) 1. 24. A. Bottino, C. Favero, N. Fornengo and G. 3. J.R. Primack, to be published in Proc. Snow- Mignola, Astroparticle Physics 3 (1995) 77. mass (eds E.W. Kolb and R. Peccei) (1995). 25. G. Jungman, M. Kamionkowski and K. Gri- 4. R.D. Peccei and H.R. Quinn, Phys. Rev. Lett. est, to be published in Phys. Rep.(1995). 38 (1977) 1440. 26. K. Griest, M. Kamionkowski and M.S. 5. E.W. Kolb and M.S. Turner, The Early Uni- Turner,Phys.Rev. D41 (1990) 3565; verse, Addison Wesley Company, 1990. M. Kamionkowski and M.S. Turner, Phys. 6. G. Raffelt, preprint hep-ph/9502358, 1995. Rev. D43 (1991) 1774. 7. B.A. Battye and E.P.S. Shellard, Nucl. Phys. 27. M. Kamionkowski, Phys. Rev. D44 (1991) 423 (1994) 260. 3021. 8. B.A. Battye and E.P.S. Shel- 28. A. Bottino, N. Fornengo, G. Mignola and L. lard, preprint DAMTP–R–94–31 (1994) and Moscoso, Astroparticle Physics 3 (1995) (65). astro-ph/9408035. th 29. J. Lopez, D. Nanopoulos and A. Zichichi, 9. C. Hagmann et al., 30 Rencontres de Phys. Lett. B291 (1992) 255. Moriond (1995) and astro–ph/9508013; 30. J. Lopez, D. Nanopoulos and H. Pois,Phys. K. Van Bibber et al., preprint UCRL–JC– Rev.D47 (1993)2468. 118357 (1994); 31. G.L. Kane, C. Kolda, L. Roszkowski and J.D. L. Cooper and G.E. Stedman, Phys. Lett. Wells, Phys. Rev. D49 (1994) 6173. B357 (1995) 464. 32. E. Diehl, G.L. Kane, C. Kolda and J.D. Wells, 10. R. Davis, Phys.Lett.B 180(1986)225 Michigan Univ. Preprint, hep-ph/9502399 11. K. Tamvakis and D. Weiler,Phys. Lett. B112 (1995). (1982) 451. 33. P. Nath and R. Arnowitt, Phys. Rev. Lett. 70 12. J.F. Nieves, Phys. Rev. D33 (1986) 1762. (1993) 3696. 13. E.J. Chun and A. Lukas, preprint hep- 34. R. Arnowitt and P. Nath, EPS conference ph/9503233 (1995). (1995) to be published. 14. T. Goto and M. Yamaguchi, Phys. Lett. B276 35. R. Rattazzi and U. Sarid, Stanford Univ. (1992) 123. preprint (1995). 15. S.A. Bonometto, F. Gabbiani and A. Masiero, 36. M.Olechowski and S.Pokorski, Phys. Lett. Phys.Lett. B139 (1989) 433. B344 (1995) 201. 16. K. Rajagopal, M.S. Turner and F. Wilczek, 37. V. Berezinsky, A. Bottino, J. Ellis, N. For- Nucl. Phys. B358 (1991) 447. nengo, G. Mignola and S. Scopel, Astroparti- 17. S.A. Bonometto, F. Gabbiani and A. Masiero, cle Physics to be published (1995). Phys.Rev. D49 (1994) 3918. 38. L.E. Ibanez and G.G. Ross, Phys. Lett. B110 18. E. Akhmedov, Z.G. Berezhiani, R.N. Moha- (1982) 215; patra and G. Senjanovic, Phys. Lett. B299 J. Ellis, D. Nanoupos and K. Tamvakis, Phys. (1993) 90. Lett. B121 (1983) 123. 19. J.Z. Rothstein, K. Babu and D. Seckel, Nucl. 39. M.Beck (Heidelberg-Moscow collaboration) Phys. B403 (1993) 725. Nucl. Phys. B (Proc.Suppl.)35 (1994) 150. 20. J.M. Cline, K. Kainulainen and K. Olive, As- troparticle Physics 1 (1993) 387. 21. V. Berezinsky and J.W.F. Valle, Phys. Lett. B318 (1993) 360.